How do you find the volume of the solid #y=sqrt(9-x^2)# revolved about the x-axis?

Answer 1

Volume #= 36pi \ "unit"^3#

graph{(y-sqrt(9-x^2))=0 [-6, 6, -2, 4]}

The Volume of Revolution about #Ox# is given by:
# V= int_(x=a)^(x=b) \ pi y^2 \ dx #
So for for this problem, Noting that #9-x^2=0 => x=+-3#, and that by symmetry we can double the volume for the region #x in [0,3]#
# V= 2int_0^3 \ pi (sqrt(9-x^2))^2 \ dx # # \ \ \= 2pi \ int_0^3 \ (9-x^2) \ dx # # \ \ \= 2pi \ [9x-x^3/3]_0^3# # \ \ \= 2pi \ {(27-27/3) - (0-0)}# # \ \ \= 36pi#
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Answer 2

To find the volume of the solid generated by revolving the curve ( y = \sqrt{9 - x^2} ) about the x-axis, you can use the method of cylindrical shells or the disk/washer method.

Using the disk/washer method, you integrate from ( x = -3 ) to ( x = 3 ) (since ( y = \sqrt{9 - x^2} ) is defined for ( -3 \leq x \leq 3 )) and evaluate the integral:

[ V = \pi \int_{-3}^{3} (\sqrt{9 - x^2})^2 , dx ]

This integral represents the volume of infinitely thin disks (or washers) at each value of ( x ), summed together over the interval ( [-3, 3] ).

After integrating, you will find the volume of the solid generated by revolving ( y = \sqrt{9 - x^2} ) about the x-axis.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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